Theorems · Theorem · group theory
CoxeterSystem.submonoid_closure_range_simple
∀ {B : Type u_1} {W : Type u_3} [inst : Group W] {M : CoxeterMatrix B} (cs : CoxeterSystem M W),
Submonoid.closure (Set.range cs.simple) = ⊤The simple reflections of W generate W as a monoid.
- Defined in
- Mathlib.GroupTheory.Coxeter.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Group
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- Top.topstatement and proof · cited by 9,680
- Groupstatement and proof · cited by 6,238
- Set.rangestatement and proof · cited by 4,705
- Subgroupproof · cited by 3,593
- Submonoidstatement and proof · cited by 3,086
- Submonoid.closurestatement and proof · cited by 167
- CoxeterMatrixstatement and proof · cited by 140
- CoxeterSystemstatement and proof · cited by 126
- Subgroup.toSubmonoidproof · cited by 114
- CoxeterSystem.simplestatement and proof · cited by 73
- Set.union_selfproof · cited by 32
Cited by4
Results whose statement or proof uses this declaration.
- CoxeterSystem.simple_induction_leftproof · cited by 1
- CoxeterSystem.ext_simpleproof · cited by 1
- CoxeterSystem.simple_inductionproof · cited by 0
- CoxeterSystem.simple_induction_rightproof · cited by 0